arXiv subjects
Publications and source records attributed to Leonid Kontorovich.
We prove an elementary yet useful inequality bounding the maximal value of certain linear programs. This leads directly to a bound on the martingale difference for arbitrarily dependent random variables, providing a generalization of some recent concentration of measure results. The linear programming inequality may be of independent interest.
We derive sufficient conditions for a family $(X^n,ρ_n,P_n)$ of metric probability spaces to have the measure concentration property. Specifically, if the sequence $\{P_n\}$ of probability measures satisfies a strong mixing condition (which we call $η$-mixing) and the sequence of metrics $\{ρ_n\}$ is what we call $Ψ$-dominated, we show that $(X^n,ρ_n,P_n)$ is a normal Levy family. We establish these properties for some metric probability spaces, including the possibly novel $X=[0,1]$, $ρ_n=\ell_1$ case.
We prove what appears to be the first concentration of measure result for hidden Markov processes. Our bound is stated in terms of the contraction coefficients of the underlying Markov process, and strictly generalizes the Markov process concentration results of Marton (1996) and Samson (2000). Somewhat surprisingly, the bound turns out to be the same as for ordinary Markov processes; this property, however, fails for general hidden/observed process pairs.
We prove an apparently novel concentration of measure result for Markov tree processes. The bound we derive reduces to the known bounds for Markov processes when the tree is a chain, thus strictly generalizing the known Markov process concentration results. We employ several techniques of potential independent interest, especially for obtaining similar results for more general directed acyclic graphical models.